Final answer:
The integral of (x² / (4x³ + 3)) dx simplifies to (1/4) ln|4x + 3| + C after using u-substitution.
Step-by-step explanation:
The student has asked to evaluate the indefinite integral ∫ (x² / (4x³ + 3)) dx. To solve this integral, we need to simplify the integrand first. The x² term in the numerator can be cancelled out by one x term in the denominator, simplifying the integrand to (1 / (4x + 3)). Now, the integral becomes ∫ (1 / (4x + 3)) dx, which is a straightforward integration problem. To integrate, we can use a u-substitution where u = 4x + 3 and du = 4 dx. With these substitutions, the integral simplifies to (1/4) ∫ (1/u) du, which evaluates to (1/4) ln|u| + C, where C is the constant of integration. Substituting back for u, we get the final answer: (1/4) ln|4x + 3| + C.